NORTH LAKHIMPUR COLLEGE, ASSAM (AUTONOMOUS)
Estd. 1952
COLLEGE WITH POTENTIAL FOR EXCELLENCE
[Re-Accredited ‘A’ Grade by NAAC with 3.08 CGPA]
SYLLABI
FOR THREE YEAR DEGREE COURSE
IN
MATHEMATICS
UNDER SEMESTER SYSTEM
B.A/B.Sc Elective(Pass) and Core (Major) Course Programme
North Lakhimpur College (Autonomous)
Department Of Mathematics
B.A./B.Sc. Mathematics Core (Major) Course
Course Structure
Semester / Course Code / Course Title / CreditI / CT-5-MTH-101 / Classical Algebra , Trigonometry , Vector Calculus / 5
II / CT-5-MTH-201 / Matrices, Ordinary Differential Equations and Numerical Analysis / 5
III / CT-4-MTH-301 / Analysis I : Real Analysis / 4
CT-4-MTH-302 / Co-Ordinate Geometry, Algebra-I / 4
IV / CT-3-MTH-401 / Computer Programming(C-Programming) / 3
CP-2-MTH-402 / Computer Laboratory / 2
CT-5-MTH-403 / Linear Programming, Analysis-II (Multiple Integral) / 5
V / CT-5-MTH-501 / Logic and Combinatorics , Complex Analysis / 5
CT-5-MTH-502 / Algebra- II , Number Theory / 5
CT-5-MTH-503 / Fluid Mechanics / 5
CT-5-MTH-504 / Mechanics , Integral Transformation / 5
PR-1-MTH-505 / Project -I / 1
VI / CT-5-MTH-601 / Statistics, Graph Theory, Fuzzy Set Theory / 5
CT-5-MTH-602 / Discrete Mathematics,Metric Spaces / 5
CT-5-MTH-603 / Linear Algebra, Partial Differential Equation / 5
CT-5-MTH-604 / Group (A): Topological Spaces,Functional Analysis
Group (B): Space Dynamics, Relativity / 5
PR-1-MTH-605 / Project-2 / 1
Total Credit / 70
Note:
- B. Sc. students taking Mathematics major will have to take two elective subjects. Moreover they will have to take one compulsory subject (English) and one skill based subject.
- B. A. students taking Mathematics major will have to take one elective subject. Moreover they will have to take two compulsory subjects (English and MIL) and one skill based subject.
Semester wise credit distribution for B.A. and B. Sc. Major in Mathematics is as follows:
Programme: B. Sc. (Mathematics major)
Semester / Core (Major) / Elective / Compulsory / Skill based / Total CreditI / 5 / 5X2=10 / 4 / 19
II / 5 / 5X2=10 / 4 / 19
III / 8 / 5X2=10 / 2 / 20
IV / 10 / 5X2=10 / 0(EVS) / 20
V / 21 / 21
VI / 21 / 21
Total Credit / 70 / 40 / 4 / 6 / 120
Programme: B. A. (Mathematics major)
Semester / Core (Major) / Elective / Compulsory / Skill based / Total CreditI / 5 / 5 / 4X2=8 / 18
II / 5 / 5 / 4X2=8 / 18
III / 8 / 5 / 4X2=8 / 21
IV / 10 / 5 / 3X2=6 / 21
V / 21 / 0(EVS) / 21
VI / 21 / 21
Total Credit / 70 / 20 / 22 / 8 / 120
B.A./B.Sc. Mathematics Elective(Pass) Course Syllabus
B.A./B.Sc. Mathematics Core(Major) Course Syllabus
SEMESTER – I
Mathematics Core (Major)
Title: Classical Algebra,Trigonometry,Vector Calculus
Code (Paper): CT-5-MTH-101
Credit: 5
Total Marks: 100
Lecture Hours: 80 L-4, T-1,P-0 Credit: 5
Objective :To introduce basic ideas of algebraic and analytic structures. Students can have a deeper insight of Trigonometry.Students will have an orientation towards the vectorial notations of multivariable calculi.
A.Classical Algebra Lecture Hours: 40
Unit I: Real sequences: Definition, bounds of a sequence, convergence of sequences and related theorems , limit of a sequence, Bolzano Weierstrass theorem, Definitions of limit inferior and superior with simple examples, Convergent sequences and statements of related theorems , non convergent sequences, Cauchy’s General Principle of convergence and Cauchy sequence, monotonic sequences.(Lecture Hours: 13)
Unit II: Infinite Series and its convergence: Introduction, Necessary condition for convergence, Cauchy’s general principle of convergence for series, Statements of preliminary theorems, positive series and its necessary condition of convergence, Geometric series , Comparison series ,Statements of comparison test (first and second types), Cauchy’s Root Test ,D’Alembert’s Ratio Test, , and Raabe’s Test, Leibnitz’s Test for convergence of an alternating Series. (Lecture Hours: 15)
Unit III: Theory of Polynomial equations: Definitions. Division algorithm, Remainder theorem, factor theorem and theorems on Existence of real roots (statements only) with examples, Descartes’ rule of sign., Fundamental Theorem of Algebra, Existence of complex roots, Relation between roots and coefficients and related problems, Transformation of equation, Cardon’s method of solution of cubic equation. (Lecture Hours: 12)
B.Trigonometry: Lecture Hours: 24
Unit I: De Moivre’s theorem and important deductions from De Moivre’s theorem (Lecture Hours: 9 )
Unit II: Trigonometrical and exponential functions of complex arguments. (Lecture Hours: 5)
Unit III : Gregory’s series and evaluation of π.( Lecture Hours: 5)
Unit IV: Summation of trigonometric series and hyperbolic functions. (Lecture Hours: 5)
C.Vector Calculus Lecture Hours: 16
Unit I: Ordinary derivatives of vectors, Space curves, Continuity and differentiability, Differentiation formulae, Partial derivatives of vectors and related problems, Vector differential operator del, Gradient, Directional derivative, Divergence and Curl, Laplacian operator , Vector identities and related problems. (Lecture Hours: 16)
Text Books :
[1] Mathematical Analysis; S. C. Malik and S. Arora, New age International (P) Ltd. New Delhi
[2] Higher Algebra; B. Das & S.R. Maity, Ashoke Prakashan, Calcutta.
[3] Higher Trigonometry; B.C. Das, B.N. Mukherjee, U.N. Dhur and Sons, Calcutta.
[4] Introduction to Real Analysis; Robert G Bartle, Donald R Sherbert; Wiley John and sons
[5] A text book of vector calculus; Shanti Narayan, J. N. Kapur, S. Chand and company, N. Delhi
Reference Books :
1. A Text Book of Higher Algebra; M.Ray, H. S. Sarma, S. Chand and Company, New Delhi.
2. Theory and Problems of Vector Analysis, Murray R. Spiegel, Schaum’s outline series, Mc Graw Hill Book Company.
3. Higher Algebra, Hall and Knight, Arihant Publication
SEMESTER – II
Mathematics Core (Major)
Title:Matrices, Ordinary Differential Equations and Numerical Analysis
Code (Paper):CT-5-MTH-201
Credit: 5
Total Marks: 100
Lecture Hours:80 L-4,T-1,P-0 Credit:5
Objective: To enable students to use matrix methods for solving liners equations. They will learn the basics of differential equations and also about the numerical methods of solving various types of equations
- Matrices Lecture Hours: 14
Unit I: Rank of a matrix, Elementary operations on a matrix, Determination of rank by reduction into echelon form & normal form, elementary matrices. (Lecture Hours: 6)
Unit II: Solution of homogeneous & non homogeneous linear equations, Characteristic polynomial, characteristic equation, Eigen values and Eigen vectors, Cayley-Hamilton theorem. (Lecture Hours: 8)
- Ordinary Differential equations: Lecture Hours: 28
Unit I: Exact differential equations of first order, Equations of first order higher degree, Clairaut’s form, wronskian, its properties and application. (Lecture Hours: 10)
Unit II: Linear differential equation of higher order with constant coefficients, linear homogeneous equations.(Lecture Hours: 9)
Unit III: Linear equation of second order with variable coefficients: Removal of first order derivative, Change of independent variables, Method of variation of parameters.
(Lecture Hours: 9)
- Numerical Analysis: Lecture Hours: 38
Unit I: Solution of algebraic and transcendental equation: Bisection method, Regula Falsi Method, Iteration method, Newton-Raphson method and its geometrical interpretation.
Solution of system of equations: Gauss elimination method, Gauss Seidal Method, Gauss Jordan method. (Lecture Hours: 13)
Unit II: Diagonal and horizontal difference tables, finite difference operators, Newton’s forward, backward and general interpolation formulae, Lagrange’s interpolation formula, Quadrature: Trapezoidal rule, Simpson’s quadrature (1/3 and 3/8 rule), Weddle’s rule (Lecture Hours: 15)
Unit III: Numerical solution of ODE, Picard’s and Eular’s Method (Lecture Hours: 10)
Text Books :
[1] A Text Book of Matrices; Shanti Narayan and P.K.Mittal, S. Chand and Company Ltd.
[2] Advanced Differential Equation; M D Raisinghania, S Chand Company.
[3] Numerical Analysis; Jain, Iyenger, Jain; New Age Publication
Reference Books :
1. Differential Equations; S. L. Ross, John Wiley and sons, India ,2004.
2. Numerical Analysis; G. Shanker Rao, New Age International Publisher..
3. Introductory Method of Numerical Analysis; S.S. Sastry, Prentice Hall of India Pvt. Ltd.
4. Introduction to Differential Equations, E A Condington
5. Numerical Methods, P. Kandasamy, S. Chand and Company
SEMESTER – III
Mathematics Core (Major)
Title:Analysis-I: Real Analysis
Code (Paper):CT-4-MTH-301
Credit: 4
Total Marks: 80
Lecture Hours: 64 L-4,T-0,P-0 Credit:4
Objective : To enable students to identify the analytical aspects of Mathematical concepts.
- Differential Calculus Lecture Hours: 30
Unit I: Successive differentiation, Leibnitz’s theorem, Indeterminate forms, Sub tangent, sub normal, derivative of arc length (Cartesian and polar forms), values of , angle between radius vector and tangent ,polar sub tangent and polar subnormal, curvature and radius of curvature. (Lecture Hours: 8)
Unit II: Function of one variable: Functions continuous on closed intervals, Differentiability, Darboux’s theorem, Rolle’s theorem, Lagrange mean value theorem, Cauchy’s mean value theorem, Taylor’s theorem, Taylor’s series, Maclaurin’s series. (Lecture Hours: 8)
Unit III: Partial derivatives, Euler’s theorem on homogeneous function. (Lecture Hours: 4)
Unit IV: Function of several variable : Explicit and implicit functions, continuity, partial derivatives, definition of Jacobian, partial derivatives of higher order, Young’s and Schwarz’s theorems(without proof), change of variables, Taylor’s theorem, extreme values. (Lecture Hours: 10)
B. Integral Calculus Lecture Hours: 10
Unit I: Reduction formula of the integrands sinnx, cosnx, tannx, andsinnxcosmx(Lecture Hours: 6)
Unit II: Rectification of plane curves, surface and volume of solids of revolution. (Lecture Hours: 4 )
C. Riemann integral Lecture Hours: 24
Unit I: Definitions and existence of R-integrals, inequalities of R-integrals, refinement and related theorems, Darboux’s theorem, conditions of integrability (both the forms). Integral as a limit of sum (Riemann sums) and its relationship with Darboux’s condition of integrability, some applications, integrability of continuous and monotonic functions, functions with finite and infinite number of discontinuities, related examples. (Lecture Hours: 10 )
Unit II: Primitive, fundamental theorem (1st & 2nd) of integral calculus, first mean value theorem and generalized first mean value theorem, related examples, Integration by parts & change of variable on an integral, second mean value theorem (statement only), particular case of second Mean Value theorem. (Lecture Hours: 5)
Unit III: Improper integrals: Introduction and their convergence, Statements of Comparison test, Cauchy’s test, Abel’s test, Dirichlet’s test and their applications. (Lecture Hours: 5)
Unit IV: Beta and Gamma functions and their relationship. (Lecture Hours: 4)
Text Books :
[1] Differential Calculus; B C Das and B N Mukherjee , U N Dhur & Sons , Private Ltd,
Calcutta.
[2] Mathematical Analysis; S C Malik & Savita Arora, New Age International (P) Ltd, New Delhi.
[3] Integral Calculus including Differential equations ; B C Das & B N Mukherjee, U N Dhur &
Sons Pvt. Ltd, Calcutta.
Reference Books :
1. Introduction to Real Analysis; R G Bartle and D R Sherbert, John Wiley and
Sons (Asia) Pvt.
2. Principals of Mathematical Analysis; Walter Rudin; Mc Graw Hill International.
3. Mathematical Analysis; Tom M Apostol, Narosa Publishing House.
4. Advanced Calculus, Schuam Series
SEMESTER – III
Mathematics Core (Major)
Title:Co-Ordinate Geometry, Algebra-I
Code (Paper):CT-4-MTH-302
Credit: 4
Total Marks: 80
Lecture Hours: 64 L-4,T-0,P-0 Credit:4
A. Co-ordinate GeometryLecture Hours: 34
(a) 2 - Dimension Lecture Hours: 20
Unit I: Transformation of coordinates: Translation of axes, Rotation of axes, Invariants, Removal of xy-term. (Lecture Hours: 4)
Unit II: Pair of straight lines: Pair of straight lines though origin, Angle and Bisectors of the angle between the lines given by homogenous equation of 2nd degree, Condition for the general equation of second degree to represent a pair of straight lines, Pair of intersecting straight lines, Pair of parallel straight lines.( Lecture Hours: 8)
Unit III: General Equation of second degree: Equation to the conic sections, Centre of a conic, Reduction to central and non central conic, Tangent to the conic and condition of tangency, Chord of contact, Pole and Polar, conjugate diameter, (Lecture Hours: 8)
(b) 3- Dimension Lecture Hours: 14
Unit I: Sphere, Section of a sphere by plane, Intersection of two spheres, Tangent line and tangent plane. (Lecture Hours: 7)
Unit II: Cone, Right circular cone, Tangent planes, Cylinder, Right circular cylinder.(Lecture Hours: 7)
B. Algebra- I Lecture Hours: 30
Unit I: Binary Composition, Definition and Examples of Group, Elementary properties and theorem of Group, Subgroups, Lagrange’s theorem, cyclic groups.(Lecture Hours: 15)
Unit II: Normal subgroups, Quotient groups, Homomorphisms – Isomorphisms, permutations, cyclic permutations, cycles of a permutation, disjoint permutations, Permutation Group, Cayley’s theorem. (Lecture Hours: 15)
Text Books :
1. A Text Book of Analytical Geometry of three Dimension ; P.K. Jain & K. Ahmed, Wiley
Eastern Ltd., 1994.
2. Analytical Geometry of two and three dimensions; R.M. Khan, New Central Book Agency
Calcutta.
Reference Books :
1. Text Book of Analytical Geometry of two Dimensions; P.K. Jain & K. Ahmed, Wiley
Eastern Ltd.
B.A./B.Sc. 4th Semester Mathematics Core (Major) Syllabus
Course Code: CT-3-MTH-401
Course Title:Computer Programming(C-Programming)
Total Marks : 60
Lecture Hours:48 L-3,T-0,P-0 Credit:3
Objective: Students will be able to formulate programs for various numerical methods to solve different types of problems. By Computer Laboratory, they will be exposed to the useful software like Matlab and Methematica.
Computer Programming:( C- Programming)Lecture Hours: 48
Unit I: Introduction to C-Programming: Basic programming concept, programming approach to solving problem, flowcharts, algorithm, character set, C tokens, keywords and identifiers, constants, variables, data types, declarations of variables, declaration of storage class, assigning values to variables. (Lecture Hours: 9)
Unit II: Operators and expressions: Arithmetic operators, relational operators, logical operators, assignment operators, increment and decrement operators, conditional operators, bitwise operators, arithmetic expressions, precedence of arithmetic operators, type conversions in expressions operator precedence and associativity, mathematical functions. (Lecture Hours: 9 )
Unit III: Input output operations: Reading and writing a character, formatted input and formatted output. (Lecture Hours: 5)
Unit IV: Decision Making and Branching, IF statement, IF … ELSE statement, nested IF, ELSE IF Ladder, WHILE statement, DO statement, FOR statement, Jumps in Loops. (Lecture Hours: 9)
Unit V: Arrays: One dimensional arrays, declaration of one dimensional arrays, initialization of one dimensional arrays, two dimensional arrays, initializing two dimensional arrays, multi-dimensional arrays. (Lecture Hours: 7)
Unit VI: User defined functions: Elements of user defined functions, Definition of functions, return values and their types, function calls, function declaration, category of functions, no arguments and no return values, arguments with return values, no arguments but returns a value, functions that return multiple values. (Lecture Hours: 6)
Unit: VII: Concept of C++, Transformation from C to C++.(Lecture Hours: 3)
Text Books :
[1] Programming in ANSI C; E Balagurusamy, Tata McGraw-Hill Publishing Company Ltd, New Delhi.
[2] C Programming; Jayapawan
[3] Programming in C; V. Rajaraman, Prentice Hall of India
Reference Books :
1. C- Programming; B.S. Gottfried, Tata McGraw Hill.
2. How to solve it; R.G.Dromey, Prentice Hall of India.
Course Code: CP-2-MTH-402
Course Title: Computer Laboratory
Lecture Hours: 32 L-0,T-0,P-2 Credit:2
Computer Laboratory (Practical)Lecture Hours: 32
(a) C- Programming Lecture Hours: 24
1. Temperature conversion 2. Area of triangle 3. Solution of linear equations
4. Simple and compound interest 5. Sum of series 6. Solution of quadratic equation
7. Checking of Prime numbers 8. Sum of sine, cosine and Fibonacci series,
9. Mean and standard deviation 10. Printing of a matrix
11. Matrix addition, subtraction, multiplication, transpose
12. Solution of equation by Newton – Raphson method, Bisection method.
13. Simpson’s 1/3 rule 14. Sorting of numbers (ascending and descending)
15. Computation of salary 16. Find the largest number among n numbers
17. Finding the factorial of a number 18. Finding factors of an integer. 19. Sum of digits of a number 20. Printing of numbers and characters in various forms, number tables.
(b) Matlab/Mathematica Lecture Hours: 8
Evaluation of arithmetic expression, exponential, logarithmic and trigonometric functions, computation of complex numbers, Plotting of curves (Algebraic function, trigonometric function and exponential function), Operations in matrices, Plotting of 3D curves and shapes, Solution of algebraic equation, simultaneous linear equations, Numerical solution of differential equation.
Text Books :
[1] Programming in ANSI C; E Balagurusamy, Tata McGraw-Hill Publishing Company Ltd, New Delhi.
[2] Getting started with Mat lab, A quick introduction for scientist and Engineers; Rudrapratap, Oxford university Press.
[3] Getting Started with MatLab; A quick Introduction for Scientists and Engineers; Rudra Pratap; Oxford University Press
[4] Methematica, Schuam Series
Referecne Books:
1.A Handbook on Mathematica Programming; B.C. Chetia,Dutta Publication
2. Elementary Matlab, P. D. Goswami, Kaustubh Prakashan
4th Semester
Mathematics (Core)
Course Code: CT-5-MTH-403
Total Marks : 100
Course Title:Linear Programming, Analysis-II (Multiple Integral)
Lecture Hours: 80 L-4,T-1,P-0 Credit:5
Objective: Students will be able to determine the Mathematical know-how of linear programming problems of Operations Research and also to solve those using LPP techniques. Students will also learn about multiple integral and their applications.
A. Linear Programming (LP) Lecture Hours: 45
Unit I: LP Model formulation & Graphical Method: Introduction, General structure and assumption of LP model, Mathematical formulation of a linear programming problem, Example of LP model Formulation, Feasible solution, basic solution, graphical method for the solution of a linear programming problem, convex set. (Lecture Hours: 8)
Unit II: Theory of simplex algorithm and simplex method: Standard form of an LP Problem, Simplex Algorithm, Solutions of unique optimal solution, alternative optimal solution, unbounded solution, artificial variable technique (Charnes’ M-technique, two phase method), Degeneracy.( Lecture Hours: 14)
Unit III: Duality Theory: Concept of duality, Types of primal dual problem, standard form, Rules for constructing the dual from primal, Simple and mixed type problems, Theorem on duality, Fundamental duality theorem(Statement only). (Lecture Hours: 8)
Unit IV: Transportation Problem: Definition, Transportation Table, Loops in transportation tables and their properties, Determination of an initial basic feasible solution by North West corner method, Matrix minima or least cost method and Vogel approximation method, unbalanced transportation problem, optimization by Modi method. (Lecture Hours: 15)
B. Analysis-II (Multiple Integral)Lecture Hours: 35
Unit I: Fourier series: Preliminary & other theorems, main theorem, series for even function, odd functions, half range series, Interval other than [-π, π] (Lecture Hours: 10)
Unit II: Integration over R2 : Line integrals , double integrals, double integrals over a region double integrals over a closed domain, Green’s theorem. (Lecture Hours: 12)
Unit III: Integration over R3 : Surface and surface integral, Stoke’s and Gauss’s theorems and their applications.(Lecture Hours: 13)
Text Books :
[1] Operations Research – Theory and Application; J.K.Sharma, McMillan India Ltd. N. Delhi. [2] Linear Programming and Game Theory; Dipak Chatterjee, Prentice Hall of India (P) Ltd
[3] Mathematical Analysis; S C Malik & Savita Arora, New Age International (P)Ltd, Publishers, New Delhi.
Reference Books :
1. Linear programming and Theory of Game ; P. M. Karak, New Central Book Agency(P) Ltd